13 research outputs found
Pseudo-Kleene algebras determined by rough sets
We study the pseudo-Kleene algebras of the Dedekind-MacNeille completion of
the ordered set of rough set determined by a reflexive relation. We
characterize the cases when PBZ and PBZ*-lattices can be defined on these
pseudo-Kleene algebras.Comment: 24 pages, minor update to the initial versio
Defining rough sets as core-support pairs of three-valued functions
We answer the question what properties a collection of
three-valued functions on a set must fulfill so that there exists a
quasiorder on such that the rough sets determined by coincide
with the core--support pairs of the functions in . Applying this
characterization, we give a new representation of rough sets determined by
equivalences in terms of three-valued {\L}ukasiewicz algebras of three-valued
functions.Comment: This version is accepted for publication in Approximate Reasoning
(May 2021
On the number of atoms in three-generated lattices
As the main achievement of the paper, we construct a three-generated,
2-distributive, atomless lattice that is not finitely presented. Also, the
paper contains the following three observations. First, every coatomless
three-generated lattice has at least one atom. Second, we give some sufficient
conditions implying that a three-generated lattice has at most three atoms.
Third, we present a three-generated meet-distributive lattice with four atoms.Comment: 12 pages, 3 figure
Orientation-Constrained Rectangular Layouts
We construct partitions of rectangles into smaller rectangles from an input
consisting of a planar dual graph of the layout together with restrictions on
the orientations of edges and junctions of the layout. Such an
orientation-constrained layout, if it exists, may be constructed in polynomial
time, and all orientation-constrained layouts may be listed in polynomial time
per layout.Comment: To appear at Algorithms and Data Structures Symposium, Banff, Canada,
August 2009. 12 pages, 5 figure
Going down in (semi)lattices of finite Moore families and convex geometries
International audienceIn this paper we first study the changes occuring in the posets of irreducible elements when one goes from an arbitrary Moore family (respectively, a convex geometry) to one of its lower covers in the lattice of all Moore families (respectively, in the semilattice of all convex geometries) defined on a finite set. Then, we show that the poset of all convex geometries that have the same poset of join-irreducible elements is a ranked join-semilattice, and we give an algorithm for computing it. Finally, we prove that the lattice of all ideals of a given poset P is the only convex geometry having a poset of join-irreducible elements isomorphic to P if and only if the width of P is less than 3.Dans ce texte, nous étudions d'abord les changements dans les ensembles ordonnés d'éléments irréductibles lorsqu'on passe d'une famille de Moore arbitraire (respectivement, d'une géométrie convexe) à l'une de ses couvertures inférieures dans le treillis de toutes les familles de Moore (respectivement, dans le demi-treillis des géométries convexes). Nous montrons ensuite que l'ensemble ordonné de toutes les géométries convexes ayant le même ensemble ordonné d'éléments sup-irréductibles est un demi-treillis rangé et nous donnons un algorithme pour le calculer. Enfin nous caractérisons les ensembles ordonnés P pour lesquels le treillis de leurs idéaux est l'unique géométrie convexe ayant son ensemble ordonné d'éléments sup-irréductibles isomorphe à P
Going down in (semi)lattices of finite Moore families and convex geometries
In this paper we first study the changes occuring in the posets of irreducible elements when one goes from an arbitrary Moore family (respectively, a convex geometry) to one of its lower covers in the lattice of all Moore families (respectively, in the semilattice of all convex geometries) defined on a finite set. Then, we show that the poset of all convex geometries that have the same poset of join-irreducible elements is a ranked join-semilattice, and we give an algorithm for computing it. Finally, we prove that the lattice of all ideals of a given poset P is the only convex geometry having a poset of join-irreducible elements isomorphic to P if and only if the width of P is less than 3.closure system;convex geometry;cover relation;join-irreducible;Moore family;poset of irreducible;semilattice
Groups with fix-set quasi-order
If X is a set, the fix-set quasiorder on a group of permutations of X is the quasiorder induced by containment of the fix-sets of elements of SX. Axioms for such quasiorders on groups have previously been given. We generalise these to allow non-faithful group actions, the resulting abstract quasiorders being called fix-orders. We characterise the possible fix-orders on a given group G in terms of certain families of subgroups of G. The special case in which the members of the defining family of subgroups are all normal is considered. Software is used to construct and analyse the lattices of fix-orders of many small finite groups
Bitopological Duality for Distributive Lattices and Heyting Algebras
We introduce pairwise Stone spaces as a natural bitopological generalization of Stone spaces—the duals of Boolean algebras—and show that they are exactly the bitopological duals of bounded distributive lattices. The category PStone of pairwise Stone spaces is isomorphic to the category Spec of spectral spaces and to the category Pries of Priestley spaces. In fact, the isomorphism of Spec and Pries is most naturally seen through PStone by first establishing that Pries is isomorphic to PStone, and then showing that PStone is isomorphic to Spec. We provide the bitopological and spectral descriptions of many algebraic concepts important for the study of distributive lattices. We also give new bitopological and spectral dualities for Heyting algebras, co-Heyting algebras, and bi-Heyting algebras, thus providing two new alternatives of Esakia’s duality